cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A380237 Number of sensed planar maps with n vertices and 2 faces.

Original entry on oeis.org

1, 2, 5, 14, 42, 140, 473, 1670, 5969, 21679, 79419, 293496, 1091006, 4078213, 15312150, 57721030, 218333832, 828408842, 3151769615, 12020870753, 45949957412, 176001205559, 675384194565, 2596119292840, 9994894356158, 38535398284100, 148772774499015, 575079507042663
Offset: 1

Views

Author

Andrew Howroyd, Jan 19 2025

Keywords

Comments

Also, by duality the number of sensed planar maps with n faces and 2 vertices.
The number of edges is n.

Links

Crossrefs

Column 2 of A379430.
Cf. A000346 (rooted), A380238 (achiral), A380239 (unsensed), A060404 (with a distinguished face), A103943 (with a distinguished vertex).
Cf. A000108, A210736.

Programs

  • PARI
    a(n) = {(binomial(n - 1, (n - 1)\2) + sumdiv(n, d, eulerphi(n/d)*(2^(2*d-1) - binomial(2*d-1, d)))/n)/2}
  • PARI
    seq(n)={my(c(d)=(1-sqrt(1-4*x^d + O(x*x^(n+d))))/(2*x^d)); Vec(1/(1 - x*c(2)) - 1 - sum(k=1, n, log(2 - c(k))*eulerphi(k)/k))/2}

Formula

a(n) = (A210736(n) + A060404(n))/2.
a(n) = (1/(2*n))*(n*binomial(n-1, floor((n-1)/2)) + Sum_{d|n} phi(n/d)*(2^(2*d-1) - binomial(2*d-1, d))).
G.f.: (1/2)*(1/(1 - x*C(x^2)) - 1 - Sum_{k>=1} log(1 - C(x^k)) * phi(k)/k), where C(x) is the g.f. of A000108.

A103943 Number of unrooted two-vertex n-edge maps in the plane (planar with a distinguished outside face).

Original entry on oeis.org

1, 3, 12, 48, 196, 798, 3248, 13184, 53416, 216018, 872344, 3518496, 14177528, 57080572, 229657792, 923474944, 3711572176, 14911097514, 59883185096, 240416320928, 964947251544, 3872021946532, 15533828715232, 62306843932928
Offset: 1

Views

Author

Valery A. Liskovets, Mar 17 2005

Keywords

References

  • V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.

Links

Crossrefs

Cf. A060404, A033504.

Programs

  • Mathematica
    f[n_] := (2^(2n - 1) - Binomial[2n - 1, n - 1] + Binomial[n - 1, Floor[n/2]])/2; Table[ f[n], {n, 24}] (* Robert G. Wilson v, Mar 24 2005 *)
Rest[CoefficientList[Series[1/8(-2+2/(1-4x)-1/Sqrt[1-4x]+1/Sqrt[1+4x]+2/Sqrt[-1+2/(1+2x)]-Sqrt[1+Sqrt[1-16x^2]]/Sqrt[1/2-8x^2]), {x, 0, 20}], x]] (* Benedict W. J. Irwin, Aug 13 2016 *)

Formula

2a(n) = 2^(2n-1) - binomial(2n-1, n-1) + binomial(n-1, floor(n/2)).
G.f.: 1/8*(2/q^2 -2 + 1/p - 1/q + 2*sqrt(p^2-2*x)/sqrt(q^2+2*x) - sqrt(2 + 2*p*q)/(p*q)), where p=sqrt(1+4*x) and q=sqrt(1-4*x). - Benedict W. J. Irwin, Aug 13 2016

Extensions

More terms from Robert G. Wilson v, Mar 24 2005

A380240 Array read by antidiagonals: A(n,k) is the number of sensed planar maps with n vertices and k faces including one distinguished outside face, n >= 1, k >= 1.

Original entry on oeis.org

1, 1, 2, 3, 1, 4, 12, 8, 2, 10, 48, 64, 25, 3, 26, 196, 412, 314, 78, 6, 80, 798, 2458, 2976, 1478, 270, 14, 246, 3248, 13452, 23588, 18844, 6748, 926, 34, 810, 13184, 70330, 166050, 192096, 110714, 30168, 3305, 95, 2704, 53416, 353716, 1074472, 1676668, 1397484, 613884, 132734, 11868, 280, 9252
Offset: 1

Views

Author

Andrew Howroyd, Jan 21 2025

Keywords

Comments

The number of edges is n + k - 2.

Examples

			Array begins:
==============================================================
n\k |  1    2      3      4       5       6       7      8 ...
----+---------------------------------------------------------
  1 |  1    1      2      4      10      26      80    246 ...
  2 |  1    3     12     48     196     798    3248  13184 ...
  3 |  1    8     64    412    2458   13452   70330 353716 ...
  4 |  2   25    314   2976   23588  166050 1074472 ...
  5 |  3   78   1478  18844  192096 1676668 ...
  6 |  6  270   6748 110714 1397484 ...
  7 | 14  926  30168 613884 ...
  8 | 34 3305 132734 ...
   ...

Crossrefs

Columns 1..2 are A002995, A060404.
Rows 1..2 are A003239(n-1), A103943.
Antidiagonal sums are A103937.
Cf. A269920 (rooted), A379430 (sensed with no root).

A113183 Number of unrooted two-face maps in the plane (considered up to orientation-preserving homeomorphism) with the faces of equal degree n: planar maps with a distinguished outside face.

Original entry on oeis.org

1, 1, 2, 3, 8, 18, 58, 155, 546, 1592, 5774, 17798, 65676, 210362, 785248, 2588155, 9743348, 32832290, 124416022, 426685544, 1625465732, 5654938190, 21636274202, 76171463926, 292498386900, 1040120036300, 4006388161846, 14369121494126
Offset: 1

Views

Author

Valery A. Liskovets, Oct 19 2005

Keywords

Examples

			There exist 2 maps in the plane with two triangular faces: a triangle and a map consisting of a 2-path and a loop in its middle vertex that separates both ends. Therefore a(3) = 2.

Links

Crossrefs

Cf. A000010, A060404, A113181.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, EulerPhi[#] * Binomial[n/# - 1, Floor[n/(2*#)]]^2 &] / n; Array[a, 30] (* Amiram Eldar, Aug 24 2023 *)
  • PARI
    a(n) = sumdiv(n, k, eulerphi(k)*binomial(n/k - 1, n\(2*k))^2)/n; \\ Michel Marcus, Oct 14 2015

Formula

a(n) = (1/n) Sum_{k|n} phi(k) C((n/k)-1,floor(n/(2k)))^2 where phi(k) is the Euler function A000010.
Showing 1-4 of 4 results.

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